The weak Stokes problem associated with a flow through a profile cascade in Lr-framework

Abstract

We study the weak steady Stokes problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade, in the L r -framework. The used mathematical model is based on the reduction to one spatial period, represented by a bounded 2D domain Ω. The corresponding Stokes problem is formulated by means of three types of boundary conditions: the conditions of periodicity on the "lower" and "upper" parts of the boundary, the Dirichlet boundary conditions on the "inflow" and on the profile and an artificial "do nothing"-type boundary condition on the "outflow". Under appropriate assumptions on the given data, we prove the existence and uniqueness of a weak solution in W 1,r (Ω) and its continuous dependence on the data. We explain the sense in which the "do nothing" boundary condition on the "outflow" is satisfied.